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CLICK HERE to DOWNLOAD: 22131 Downloads so far )Mathematics Syllabus IFoS Main Exam, 2021, 2022, 2023
Paper I : Mathematics for IFoS Mains Exam
Max. Marks: 200
Duration: 3 hours
Section-A
Linear Algebra
Vector, space, linear dependance and independance, subspaces, bases, dimensions.
Finite dimensional vector spaces.
Matrices, Cayley-Hamiliton theorem, eigenvalues and eigenvectors, matrix of linear
transformation, row and column reduction, Echelon form, eqivalence, congruences and
similarity, reduction to cannonical form, rank, orthogonal, symmetrical, skew
symmetrical, unitary, hermitian, skew-hermitian formstheir eigenvalues. Orthogonal
and unitary reduction of quadratic and hermitian forms, positive definite quardratic
forms.
Calculus :
Real numbers, limits, continuity, differerentiability, mean-value theorems, Taylor's
theorem with remainders, indeterminate forms, maximas and minima, asyptotes.
Functions of several variables: continuity, differentiability, partial derivatives, maxima
and minima, Lagrange's method of multipliers, Jacobian. Riemann's definition of
definite integrals, indefinite integrals, infinite and improper intergrals, beta and gamma
functions. Double and triple integrals (evaluation techniques only). Areas, surface and
volumes, centre of gravity.
Analytic Geometry :
Cartesian and polar coordinates in two and three dimesnions, second degree equations
in two and three dimensions, reduction to cannonical forms, straight lines, shortest
distance between two skew lines, plane, sphere, cone, cylinder., paraboloid, ellipsoid,
hyperboloid of one and two sheets and their properties.
Section-B
Ordinary Differential Equations :
Formulation of differential equations, order and degree, equations of first order and
first degree, integrating factor, equations of first order but not of first degree, Clariaut's
equation, singular solution.
Higher order linear equations, with constant coefficients, complementary function and
particular integral, general solution, Euler-Cauchy equation.
Second order linear equations with variable coefficients, determination of complete
solution when one solution is known, method of variation of parameters.
Dynamics, Statics and Hydrostatics :
Degree of freedom and constraints, rectilinerar motion, simple harmonic motion,
motion in a plane, projectiles, constrained motion, work and energy, conservation of
energy, motion under impulsive forces, Kepler's laws, orbits under central forces,
motion of varying mass, motion under resistance.
Equilibrium of a system of particles, work and potential energy, friction, common
catenary, principle of virtual work, stability of equilibrium, equilibrium of forces in
three dimensions.
Pressure of heavy fluids, equilibrium of fluids under given system of forces Bernoulli's
equation, centre of pressure, thrust on curved surfaces, equilibrium of floating bodies,
stability of equilibrium, metacentre, pressure of gases.
Vector Analysis :
Scalar and vector fields, triple, products, differentiation of vector function of a scalar
variable, Gradient, divergence and curl in cartesian, cylindrical and spherical
coordinates and their physical interpretations. Higher order derivatives, vector
identities and vector quations.
Application to Geometry:
Curves in space, curvature and torision. Serret-Frenet's formulae, Gauss and Stokes' theorems, Green's identities.
Paper II : Mathematics for IFoS Mains Exam
Max. Marks: 200
Duration: 3 hours
Algebra :
Groups, subgroups, normal subgroups, homomorphism of groups quotient groups basic
isomorophism theorems, Sylow's group, permutation groups, Cayley theorem. Rings
and ideals, principal ideal domains, unique factorization domains and Euclidean
domains. Field extensions, finite fields.
Real Analysis :
Real number system, ordered sets, bounds, ordered field, real number system as an
ordered field with least upper bound property, cauchy sequence, completeness,
Continuity and uniForm continuity of functions, properties of continuous functions on
compact sets. Riemann integral, improper integrals, absolute and conditional
convergence of series of real and complex terms, rearrangement of series. UniForm
convergence, continuity, differentiability and integrability for sequences and series of
functions. Differentiation of fuctions of several variables, change in the order of partial
derivatives, implict function theorem, maxima and minima. Multiple integrals.
Complex Analysis :
Analytic function, Cauchy-Riemann equations, Cauchy's theorem,
Cauchy's integral formula, power series, Taylor's series, Laurent's Series, Singularities,
Cauchy's residue theorem, contour integration. Conformal mapping, bilinear
transformations.
Linear Programming :
Linear programming problems, basic solution, basic feasible solution and optimal
solution, graphical method and Simplex method of solutions. Duality.
Transportation and assignment problems. Travelling salesman problmes.
Section-B
Partial differential equations :
Curves and surfaces in three dimesnions, formulation of partial differential equations,
solutions of equations of type dx/p=dy/q=dz/r; orthogonal trajectories, pfaffian
differential equations; partial differential equations of the first order, solution by
Cauchy's method of characteristics; Charpit's method of solutions, linear partial
differential equations of the second order with constant coefficients, equations of
vibrating string, heat equation, laplace equation.
Numerical Analysis and Computer programming :
Numerical methods :
Solution of algebraic and transcendental equations of one variable
by bisection, Regula-Falsi and Newton-Raphson methods, solution of system of linear
equations by Gaussian elimination and Gauss-Jordan (direct) methods, Gauss-
Seidel(iterative) method. Newton's (Forward and backward) and Lagrange's method of
interpolation.
Numerical integration:
Simpson's one-third rule, tranpezodial rule, Gaussian quardrature formula.
Numerical solution of ordinary differential equations: Euler and Runge Kutta-methods.
Computer Programming :
Storage of numbers in Computers, bits, bytes and words, binary system. arithmetic and logical operations on numbers. Bitwise operations. AND,
OR , XOR, NOT, and shift/rotate operators. Octal and Hexadecimal Systems.
Conversion to and Form decimal Systems.
Representation of unsigned integers, signed integers and reals, double precision reals
and long integrers.
Algorithms and flow charts for solving numerical analysis problems.
Developing simple programs in Basic for problems involving techniques covered in the
numerical analysis.
Mechanics and Fluid Dynamics :
Generalised coordinates, constraints, holonomic and non-holonomic , systems. D'
Alembert's principle and Lagrange' equations, Hamilton equations, moment of intertia,
motion of rigid bodies in two dimensions.
Equation of continuity, Euler's equation of motion for inviscid flow, stream-lines, path
of a particle, potential flow, two-dimensional and axisymetric motion, sources and
sinks, vortex motion, flow past a cylinder and a sphere, method of images. Navier-
Stokes equation for a viscous fluid.
Recommended Textbooks:
Linear Algebra - K.C. Prasad, K B Datta
Calculus - Santhi Narayan
Analytic Geometry - Shantinarayan
Dynamics, Statistics and Hydrostatics - M.Ray
Vector analysis - Shantinarayan
Paper II
Algebra - K C Prasad, KB Datta
Real Analysis - Shantinarayan,Royden
Complex Analysis - GK Ranganath
Linear Programming - SD sharma
Numerical analysis and Computer Programming - V. Rajaraman
Mechanics & Fluid dynamics - AP Mathur
Other Books
Modern Algebra - A. R. Vashisth
Algebra - J. N. Sharma and S. N. Goyal
Linear Algebra - Sharma and Vashisth
Matrices - Vashisth
Calculus - I. A. Menon
Coordinate Geometry - R. L. Soni
Differential Calculus - Gorakh Prasad
Analytical Solid Geometry - ShantiNarayanan
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